Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2018 Higgs Boson Pair Production via Gluon Fusion: Full NLO QCD Corrections Glaus, Seraina Abstract: The measured properties of the particle detected six years ago at the LHC at CERN indicate that it is compatible with the Higgs boson predicted by the Standard Model. However, the theoretical and experimental uncertainties allow associations with extended models. Therefore it is of essential importance to investigate the properties of this particle in more detail. The determination of the Higgs potential is crucial to test whether this particle causes electroweak symmetry breaking. The self-coupling strength has to be determined to measure the Higgs potential. This can be achieved in a first step by measuring the trilinear self-coupling in Higgs pair production. At the LHC, the dominant process of Higgs pair production is the loop induced gluon fusion. Therefore the main goal of this thesis is the calculation of the next-to-leading order (NLO) QCD corrections considering the complete top-quark mass dependence in the framework of the Standard Model. The relevant two-loop integrals cannot be calculated analytically with the currently known methods. Instead a numerical integration is required. The main challenge is the extraction of the ultraviolet, the infrared and the collinear divergences from the amplitudes. For this purpose, a modified end-point subtraction has been developed for the extraction of the IR-singularities. Publicly available programs have been used for the real corrections. The differential cross section has been obtained as a distribution in the invariant Higgs pair mass. It shows that the main contributions to the cross section emerge from the invariant Higgs pair masses between 300 and 800 GeV and that the heavy-top limit is a reasonable approximation for invariant Higgs pair masses only up to about 600 GeV. Further, it can be observed that for an invariant Higgs pair mass up to 400-600 GeV the NLO QCD corrections can be reasonably approximated by the K-factor of the triangular contributions alone. The obtained hadronic cross section implies a negative contribution of about −15% from NLO mass effects compared to the previous known heavy-top limit results involving the full leading-order mass dependence. Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-169490 Dissertation Published Version Originally published at: Glaus, Seraina. Higgs Boson Pair Production via Gluon Fusion: Full NLO QCD Corrections. 2018, University of Zurich, Faculty of Science. Higgs Boson Pair Production via Gluon Fusion: Full NLO QCD Corrections Dissertation zur Erlangung der naturwissenschaftlichen Doktorwürde (Dr. sc. nat.) vorgelegt der Mathematisch-naturwissenschaftlichen Fakultät der Universität Zürich von Seraina Glaus aus Thalwil ZH Promotionskommission Prof. Dr. Thomas Gehrmann (Vorsitz) PD Dr. Michael Spira (Leitung der Dissertation) Prof. Dr. Massimiliano Grazzini Prof. Dr. Adrian Signer Zürich, 2018 Abstract The measured properties of the particle detected six years ago at the LHC at CERN indicate that it is compatible with the Higgs boson predicted by the Standard Model. However, the theoretical and experimental un- certainties allow associations with extended models. Therefore it is of essential importance to investigate the properties of this particle in more detail. The determination of the Higgs potential is crucial totestwhether this particle causes electroweak symmetry breaking. The self-coupling strength has to be determined to measure the Higgs potential. This can be achieved in a first step by measuring the trilinear self-coupling in Higgs pair production. At the LHC, the dominant process of Higgs pair produc- tion is the loop induced gluon fusion. Therefore the main goalofthisthe- sis is the calculation of the next-to-leading order (NLO) QCD corrections considering the complete top-quark mass dependence in the framework of the Standard Model. The relevant two-loop integrals cannot be calcu- lated analytically with the currently known methods. Instead a numerical integration is required. The main challenge is the extraction of the ultra- violet, the infrared and the collinear divergences from the amplitudes. For this purpose, a modified end-point subtraction has been developed for the extraction of the IR-singularities. Publicly available programs have been used for the real corrections. The differential cross sectionhasbeenob- tained as a distribution in the invariant Higgs pair mass. It shows that the main contributions to the cross section emerge from the invariant Higgs pair masses between 300 and 800 GeV and that the heavy-top limit is a reasonable approximation for invariant Higgs pair masses only up to about 600 GeV. Further, it can be observed that for an invariant Higgs pair mass up to 400-600 GeV the NLO QCD corrections can be reasonably approximated by the K-factor of the triangular contributions alone. The obtained hadronic cross section implies a negative contribution of about 15% from NLO mass effects compared to the previous known heavy-top − limit results involving the full leading-order mass dependence. Zusammenfassung Die gemessenen Eigenschaften des vor sechs Jahren am CERN detek- tierten Teilchens weisen darauf hin, dass es sich um das langgesuchte Higgsboson des Standardmodells handelt. Jedoch lassen die theoretis- chen wie auch experimentellen Unsicherheiten Zuordnungen zuanderen erweiterten Modellen zu. Deshalb ist es von ausschlaggebender Bedeu- tung, die Eigenschaften des detektierten Teilchens genauerzubestim- men. Dabei spielt die Messung des Higgspotentials eine zentrale Rolle, um zu pr¨ufen, ob dieses Teilchen effektiv f¨ur die elektroschwache Symme- triebrechung verantwortlich ist. Um das Higgspotential zu messen, muss die Selbstwechselwirkungsst¨arke zwischen Higgsbosonen direkt bestimmt werden. Dies ist ¨uber Higgspaarproduktion zur Bestimmung der trilin- eare Kopplung m¨oglich. Der dominante Prozess der Higgspaarproduktion ist die loop-induzierte Gluonfusion. Ziel dieser Arbeit ist die Berechnung der NLO QCD-Korrekturen unter Ber¨ucksichtigung der vollen Topquark- massenabh¨angigkeit im Rahmen des Standardmodells. Die relevanten Zwei-Loop-Integrale k¨onnen analytisch nicht mit gegenw¨artig bekannten Methoden berechnet werden. Stattdessen ist eine numerischeIntegration n¨otig. Die gr¨osste Herausforderung ist die Extraktion der ultraviolet- ten, der infraroten wie auch der kollinearen Divergenzen der Amplitu- den. Dazu wurde die bekannte Endpunktsubtraktionsmethode erweitert. F¨ur die reellen Korrekturen wurden die Programme FeynArts und Form- Calc verwendet. Der differentielle Wirkungsquerschnitt wurde als Funk- tion der invarianten Higgspaarmasse bestimmt. Daraus wird ersichtlich, dass die gr¨ossten Beitr¨age zum Wirkungsquerschnitt von dem invarianten Higgspaarmassenbereich zwischen 300 und 800 GeV kommen, und dass der Heavy-Top-Limit bis ungef¨ahr 600 GeV eine gute N¨aherung ist. F¨ur eine invariante Higgspaarmasse bis 400-600 GeV k¨onnen die NLOQCD- Korrekturen durch den K-Faktor der Dreiecksbeitr¨ageangen¨ahert werden. Der berechnete totale hadronische Wirkungsquerschnitt impliziert NLO Masseneffekte von 15% verglichen zu den bekannten Heavy-Top-Limit − Resultaten, welche die volle Massenabh¨angigkeit in f¨uhrender Ordnung ber¨ucksichtigen. Contents 1Introduction 3 1.1 Standard Model . 3 1.2 Motivation . 9 1.3 Previous Work . 12 1.4 Topic of this Thesis . 13 2Higgspairproduction 15 2.1 Leading-Order Cross Section . 18 2.2 VirtualCorrections ............................ 24 2.2.1 Box Diagrams . 25 2.2.2 One-Particle Reducible Diagrams . 34 2.2.3 Triangular Diagrams . 35 2.2.4 Renormalization . 36 2.2.5 FinitePartoftheVirtualCorrections . 39 2.2.6 NumericalEvaluation. 40 2.3 Real Corrections . 45 2.3.1 TheHeavy-TopLimit. 45 2.3.2 Massive Calculation . 51 3 Results and Discussion 55 3.1 Differential Cross Section . 55 3.2 TotalHadronicCrossSection . 58 4Conclusions 63 AVirtualcontributions 65 BRealcorrections 69 Bibliography 72 1 2 Chapter 1 Introduction 1.1 Standard Model The Standard Model (SM) of particle physics is a successful and elegant model to describe particle physics processes in nature. The SM provides a set of elementary particles and characterizes their properties and interactions in the framework of a spontaneously broken gauge theory. Its development is basedonexperimentaldata gathered in collider and low-energy experiments. The SM is a quantum field theory which is symmetric under the non-abelian SU(3) SU(2) U(1) gauge group. C ⊗ L ⊗ Y The first one is the symmetry group of the strong interaction between quarks and gluons described by quantum chromodynamics (QCD) [1–9]. The SU(2) U(1) L ⊗ Y symmetry describes the unified electromagnetic and weak interactions [10–12]. Leptons and quarks build together the fundamental constituents of matter charac- terized by their quantum numbers. For every lepton and quark there is an antiparticle with the same mass but opposite inner quantum numbers. Leptons and quarks appear in three generations with left-handed isospin doublets and right-handed singlets. ν ν ν e e µ µ τ τ e R µ R τ R ! "L ! "L ! "L (1.1) u uR c cR t tR d d s s b b ! "L R ! "L R ! "L R The Lagrange density of the SM, LSM,canbesplitintothekineticpartofthegauge bosons, LV , the kinetic properties of the fermions and their couplings to the gauge bosons, Lf , the Higgs sector, LH , and the Yukawa interaction of the Higgs boson with the fermions, LY , LSM = LV + Lf + LH + LY . (1.2) 3 Every generator of the symmetry group results in a gauge field. Hence we have 8 a a gluon fields Gµ (a=1,...,8) for the strong interaction and 4 fields Wµ (a=1,...3) and Bµ for the electroweak interaction. The kinetic part of the gauge fields can then be written as 1 1 V V 2 LV = Tr Dµ , Dν (1.3) −2 igV V=G,W,B # $ % V a a a with the covariant derivatives Dµ = ∂µ + igV Vµ and the gauge fields Vµ = T Vµ . T are the generators of the associated symmetry group and gV the coupling strength of the corresponding gauge field.